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Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

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Some Aspects of Hankel Matrices in Coding Theory and Combinatorics Some Aspects of Hankel Matrices in Coding Theory and Combinatorics Ulrich Tamm Department of Computer Science University of Chemnitz 09107 Chemnitz, Germany [email protected] Submitted: December 8, 2000; Accepted: May 26, 2001. MR Subject Classifications: primary 05A15, secondary 15A15, 94B35 Abstract Hankel matrices consisting of Catalan numbers have been analyzed by various authors. Desainte- Q i+j+2n Catherine and Viennot found their determinant to be 1≤i≤j≤k i+j and related them to the Q i+j−1+2n Bender - Knuth conjecture. The similar determinant formula 1≤i≤j≤k i+j−1 can be shown 2m+1 to hold for Hankel matrices whose entries are successive middle binomial coefficients m . Generalizing the Catalan numbers in a different direction, it can be shown that determinants of 1 3m+1 Hankel matrices consisting of numbers 3m+1 m yield an alternate expression of two Mills { Robbins { Rumsey determinants important in the enumeration of plane partitions and alternat- ing sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan { like numbers. The well - known relation of Hankel matrices to orthogonal polyno- mials further yields a combinatorial application of the famous Berlekamp { Massey algorithm in Coding Theory, which can be applied in order to calculate the coefficients in the three { term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices. I. Introduction A Hankel matrix (or persymmetric matrix) c0 c1 c2 ... cn−1 c1 c2 c3 ... cn c c c ... c An = 2 3 4 n+1 . (1.1) . . cn−1 cn cn+1 ... c2n−2 is a matrix (aij) in which for every r the entries on the diagonal i + j = r are the same, i.e., ai;r−i = cr for some cr. the electronic journal of combinatorics 8 2001, #A1 1 For a sequence c0,c1,c2,... of real numbers we also consider the collection of Hankel (k) matrices An , k =0, 1,..., n =1, 2,...,where ck ck+1 ck+2 ... ck+n−1 ck+1 ck+2 ck+3 ... ck+n (k) c c c ... c An = k+2 k+3 k+4 k+n+1 . (1.2) . . ck+n−1 ck+n ck+n+1 ... ck+2n−2 So the parameter n denotes the size of the matrix and the 2n − 1 successive elements ck,ck+1,...,ck+2n−2 occur in the diagonals of the Hankel matrix. We shall further denote the determinant of a Hankel matrix (1.2) by (k) (k) dn =det(An ). (1.3) Hankel matrices have important applications, for instance, in the theory of moments, and in Pad´e approximation. In Coding Theory, they occur in the Berlekamp - Massey algorithm for the decoding of BCH - codes. Their connection to orthogonal polynomials often yields useful applications in Combinatorics: as shown by Viennot [76] Hankel deter- minants enumerate certain families of weighted paths, Catalan – like numbers as defined by Aigner [2] via Hankel determinants often yield sequences important in combinatorial enumeration, and as a recent application, they turned out to be an important tool in the proof of the refined alternating sign matrix conjecture. The framework for studying combinatorial applications of Hankel matrices and further aspects of orthogonal polynomials was set up by Viennot [76]. Of special interest are 1 2m+1 determinants of Hankel matrices consisting of Catalan numbers 2m+1 m .Desainte– (k) Catherine and Viennot [24] provided a formula for det(An )andalln ≥ 1, k ≥ 0incase c that the entries m are Catalan numbers, namely c 1 2m+1 m , ,... For the sequence m = 2m+1 m , =0 1 of Catalan numbers it is Y i j n d(0) d(1) ,d(k) + +2 k ≥ ,n≥ . n = n =1 n = i j for 2 1 (1 4) 1≤i≤j≤k−1 + Desainte–Catherine and Viennot [24] also gave a combinatorial interpretation of this de- terminant in terms of special disjoint lattice paths and applications to the enumeration of Young tableaux, matchings, etc.. Q i+j−1+c c They studied (1.4) as a companion formula for 1≤i≤j≤k i+j−1 , which for integer was shown by Gordon (cf. [67]) to be the generating function for certain Young tableaux. For even c =2n this latter formula also can be expressed as a Hankel determinant formed 2m+1 of successive binomial coefficients m . c 2m+1 m , ,... For the binomial coefficients m = m , =0 1 Y i j − n d(0) ,d(k) + 1+2 k, n ≥ . n =1 n = i j − for 1 (1 5) 1≤i≤j≤k + 1 the electronic journal of combinatorics 8 2001, #A1 2 We are going to derive the identities (1.4) and (1.5) simultaneously in the next section. Our main interest, however, concerns a further generalization of the Catalan numbers and their combinatorial interpretations. In Section III we shall study Hankel matrices whose entries are defined as generalized c 1 3m+1 Catalan numbers m = 3m+1 m . In this case we could show that nY−1 j j j Yn 6j−2 d(0) (3 + 1)(6 )!(2 )!,d(1) 2j . n = j j n = 4j−1 (1 6) j=0 (4 + 1)!(4 )! j=1 2 2j These numbers are of special interest, since they coincide with two Mills – Robbins – Rum- sey determinants, which occur in the enumeration of cyclically symmetric plane partitions and alternating sign matrices which are invariant under a reflection about a vertical axis. The relation between Hankel matrices and alternating sign matrices will be discussed in Section IV. Let us recall some properties of Hankel matrices. Of special importance is the equation c0 c1 c2 ... cn−1 an;0 −cn c1 c2 c3 ... cn an;1 −cn+1 c2 c3 c4 ... cn+1 · an;2 = −cn+2 . (1.7) . . . . . . cn−1 cn cn+1 ... c2n−2 an;n−1 −c2n−1 (0) It is known (cf. [16], p. 246) that, if the matrices An are nonsingular for all n, then the polynomials j j−1 j−2 tj(x):=x + aj;j−1x + aj;j−2x + ...aj;1x + aj;0 (1.8) form a sequence of monic orthogonal polynomials with respect to the linear operator T l l mapping x to its moment T (x )=cl for all l,i.e. T (tj(x) · tm(x)) = 0 for j =6 m. (1.9) and that m T (x · tj(x))=0form =0,...,j− 1. (1.10) In Section V we shall study matrices Ln =(l(m, j))m;j=0;1;:::;n−1 defined by m l(m, j)=T (x · tj(x)) (1.11) By (1.10) these matrices are lower triangular. The recursion for Catalan – like numbers, as defined by Aigner [2] yielding another generalization of Catalan numbers, can be derived via matrices Ln with determinant 1. Further, the Lanczos algorithm as discussed in [13] t yields a factorization Ln = An · Un,whereAn is a nonsingular Hankel matrix as in (1.1), Ln is defined by (1.11) and the electronic journal of combinatorics 8 2001, #A1 3 100... 00 a1;0 10... 00 a a ... Un = 2;0 2;1 1 00 . (1.12) . . an−1;0 an−1;1 an−2;2 ... an−1;n−2 1 is the triangular matrix whose entries are the coefficients of the polynomials tj(x), j = 0,...,n− 1. In Section V we further shall discuss the Berlekamp – Massey algorithm for the decoding of BCH–codes, where Hankel matrices of syndromes resulting after the transmission of a code word over a noisy channel have to be studied. Via the matrix Ln defined by (1.11) it will be shown that the Berlekamp – Massey algorithm applied to Hankel matrices with real entries can be used to compute the coefficients in the corresponding orthogonal polynomials and the three – term recurrence defining these polynomials. Several methods to find Hankel determinants are presented in [61]. We shall mainly concentrate on their occurrence in the theory of continued fractions and orthogonal poly- nomials. If not mentioned otherwise, we shall always assume that all Hankel matrices An under consideration are nonsingular. Hankel matrices come into play when the power series 2 F (x)=c0 + c1x + c2x + ... (1.13) (0) (1) is expressed as a continued fraction. If the Hankel determinants dn and dn are different from 0 for all n the so–called S–fraction expansion of 1 − xF (x) has the form c0x 1 − xF (x)=1− q x . (1.14) − 1 1 e1x 1 − q x − 2 1 e2x 1 − 1 − ... (k) Namely, then (cf. [55], p. 304 or [78], p. 200) for n ≥ 1 and with the convention d0 =1 for all k it is d(1) · d(0) d(0) · d(1) q n n−1 ,e n+1 n−1 . n = (1) (0) n = (0) (1) (1 15) dn−1 · dn dn · dn For the notion of S– and J– fraction (S stands for Stieltjes, J for Jacobi) we refer to the standard books by Perron [55] and Wall [78]. We follow here mainly the (qn,en)–notation of Rutishauser [65]. 1 For many purposes it is more convenient to consider the variable x in (1.13) and study power series of the form the electronic journal of combinatorics 8 2001, #A1 4 c c c 1 F 1 0 1 2 ... x (x)= x + x2 + x3 + (1 16) and its continued S–fraction expansion c0 q1 x − e − 1 1 q2 x − e − 2 1 x − ... which can be transformed to the J–fraction c0 (1.17) β1 x − α1 − β2 x − α2 − β3 x − α3 − x − α4 − ..
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